Euclidean Spanners: Short, Thin, and Lanky
Sunil Arya* Gautam Dast David M. Mount~ Jeffrey S. Salowe$ Michiel Smid”
Abstract
Euclidean spanners are important data structures
in geometric algorithm design, because they pro-
vide a means of approximating the complete Eu-
clidean graph with only O(n) edges, so that the
shortest path length between each pair of points is
not more than a constant factor longer than the
Euclidean distance between the points. In many
applications of spanners, it is important that the
spanner possess a number of additional properties:
low tot al edge weight, bounded degree, and low
diameter. Existing research on spanners has con-
sidered one property or the other. We show that it
is possible to build spanners in optimal O (n log n)
time and O(n) space that achieve optimal or near
optimal tradeoffs between all combinations of these
*Max-Planck-Institut fiir Informatik, D-66123 Saarbruc-
ken, Germany. Email: {arya, michiel}@mpi-sb. mpg. de.
Supported by the ESPRIT Basic Research Actions Program,
under contract No. 7141 (project ALCOM 11).
t Math Sciences Dept., The University of Memphis, Mem-
phis, TN 38152. Supported in part by NSF Grant CCR-
9306822. E-mail: dasg@next 1.msci .memst . edu.
i Department of Computer Science and Institute for Ad-
vanced Computer Studies, University of Maryland, Col-lege Park, Maryland. Partially supported by NSF GrantCCR-93107O5. This work was done while visiting theMax-Planck-Institut fiir Informatik, Saarbriicken. E-mail:mount @cs. umd. edu.
SQue~Tech, IIIC., 7600A Leesburg Pike, Falls Church,
VA 22043. This work was done while visiting the
Max-Planck-Institut fiir Informatik, Saarbriicken. E-mail:
jsalowet!nvl, army .mil.
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properties. We achieve these results in large part
because of a new structure, called the dumbbell tree
which provides a method of decomposing a spanner
into a constant number of trees, so that each of the
O(n2) spanner paths is mapped entirely to a path
in one of these trees.
1 Introduction
Let G = (V, E) be a weighted graph, and let
dG(u, v) be the length of a shortest path between
vertices u and v in G. Let t > 1 be any constant. A
subgraph G’ is a t-spanner for G if, for all pairs of
vertices u and v, dGl(u, v)/dG(u, v) < t. When V
is a set of n points in IRk, G is the complete graph,
and the length of edge (u, v) is the Euclidean dis-
t ante between these points, we call. G a complete
Euclidean graph and Gt a Euclidean t-spanner. For
the purposes of deriving asymptotic bounds, we as-
sume that the dimension k and the spanner factor t
are constants independent of n. It is known how to
construct a Euclidean t-spanner having O(n) edges
in O(nlogn) time [5, 13, 14].
Spanners are important geometrical structures,
since they provide a mechanism for approximating
the complete Euclidean graph in a much more eco-
nomical form. Of course, a spanner should have a
small number of edges (ideally O(n)), but for many
applications, it is quite important that the spanner
be endowed with other properties. These include
the following:
Low weight: The total sum of the edge lengths
in the spanner should be as smlall as possible.
The best that can be hoped for is some con-
stant times the weight of the minimum span-
ning tree, O(w(MST)).
489
Bounded degree: The number of edges incident
to any vertex should be bounded.
Small spanner diameter: The spanner diame-
ter (or simply diameter) is defined as the
smallest integer D such that for any pair of
vertices, u and v, there is a t-spanner path
between u and v containing at most D edges.
For spanners of bounded degree the best that
can be hoped for is logarithmic diameter. In
some applications even smaller diameters may
be desirable, but this comes at the expense of
increasing degree.
A natural analogy can be made between span-
ners and a transportation network of roads con-
necting a large number of locations. Low weight
means that the amount of concrete needed to build
the roads is small, bounded degree means that no
location in the network has more than a bounded
number of roads incident to it, and small diame-
ter means that it is possible to describe any span-
ner path concisely. Existing work on spanners has
focused on achieving one property or the other.
However, a transportation network which achieves
small diameter by massively increasing total weight
is of little practical value. This suggests the impor-
tant question of whether there exist spanners that
simultaneously achieve some or all of these proper-
ties.
In this paper we present a strong positive answer
to this question. We present a number of new con-
structions for spanners. In almost all cases these
constructions are provably optimal from the per-
spectives of computation time, space, and perfor-
mance on the properties listed above. The prob-
lem is complicated by the fact that there are ob-
vious tradeoffs between these properties. (For ex-
ample, reducing diameter requires the creation of
long edges, which in turn increases total weight, or
may increase the number of edges needed in the
spanner. ) For this reason, we consider all possible
combinations of these properties.
The results of this paper arise from a number
of improved techniques in spanner constructions,
but one deserves particular mention. An important
data structure used in the construction of span-
ners is the well-separated pair decomposition, in-
troduced by Callahan and Kosaraju [4]. This struc-
ture represents the 0(n2) pairs of points using only
O(n) pairs of geometrically “well-separated” pairs
of subsets of points (definitions will be given later).
In this paper, we present a novel method of further
decomposing a well-separated pair decomposition
into a constant number of hierarchically organized
sets of well-separated pairs. (The constant depends
on the dimension and the separation factor. ) Using
this decomposition, we show that a class of span-
ners can be viewed as being the union of a con-
stant number of trees, which we call dumbbell trees.
Moreover, each of the O (nz) spanner paths arises as
the unique path between two leaves in one of these
trees. The fact that the O (n2 ) spanner paths can
be partitioned among a constant number of trees
is a rather remarkable fact in itself, and suggests a
great deal about special structure of these graphs.
Because of the importance of well-separated pair
decompositions to a variety of geometric problems,
we suspect that this decomposition may be of use to
other geometric problems. The idea of dumbbells
has appeared before [7], but their use in decompos-
ing spanner paths is new to this paper.
Here is a summary of the results in this paper.
All of the spanner constructions described below
run in optimal O (n log n) time and O(n) space for
any fixed dimension k.
Degree: We present an optimal O(n log n) time
construction for spanners of bounded degree.
This improves the best known algorithm,
due to Arya and Smid [3], which runs in
O(nlog~ n) time.
Weight: We present an optimal O(n log n) time
spanner construction that has optimal weight
O(W(MST)). This improves the best known
construction for spanners of low weight, which
was due to Das and Narasimhan [8], and which
runs in O (n log2 n) time.
Diameter: Arya, Mount and Smid [2] give ran-
domized and deterministic constructions of
spanners with O(n) edges and O(log n) span-
ner diameter. We show that it is possible to
achieve diameter a(n) + 2 with the same num-
ber of edges, where a(n) is the inverse of Ack-
ermann’s function. Furthermore, we present
a spectrum of tradeoffs between size and di-
amet er. For example, we construct spanners
of diameter 2 with O (n log n) edges, diameter
490
3 with O(n log log n) edges, diameter 4 with
O(n log* n) edges, and so on. All these span-
ners have an optimal number of edges for the
given diameter.
Degree and weight: The low-weight construc-
tion mentioned already has bounded degree,
and hence provides an optimal solution to this
problem as well. There are no previous results
on this problem.
Weight and diameter: By using a new analy-
sis tool, we show that the deterministic low-
diameter construction of Arya, Mount and
Smid [2] has weight O(W(MST) log n) as well
as diameter O (log n). This combination is op-
timal. No simultaneous bounds were previ-
ously known.
Degree and diameter: We show how to con-
struct a spanner with bounded degree and
O(log n) diameter. This is optimal with re-
spect to both diameter bound and construc-
tion time. No simultaneous bounds were
known for this problem.
Degree, weight and diameter: We show how
to construct a spanner with bounded de-
gree, weight O ( w(JfSZ’) log2 n), and diameter
O(log n). No simultaneous bounds were previ-
ously known.
In summary, all of our results are optimal in terms
of providing the best tradeoffs between these prop-
erties, except for the spanner having simultane-
ously bounded degree, low weight, and low diam-
eter, which is possibly suboptimal by at most an
O(log n) factor in weight.
The rest of this paper is organized as follows. In
Section 2, we briefly recall the well-separated pair
decomposition. In Section 3, we define the dumb-
bell tree, and show that there exists a spanner that
can be decomposed into a constant number of such
trees. In Section 4, we give a simple optimal algo-
rithm for constructing a t-spanner of bounded de-
gree. In Section 5, we show that the spanner that
results from the well-separated pair decomposition
can be pruned in such a way that we get a spanner
of weight O(W(MST)). Section 6 considers span-
ners of low diameter. Our results of Section 3 imply
that it suffices to add edges to a constant number
of bounded degree trees in order to get a spanner
of low diameter. This is done by using a technique
due to Alon and Schieber[l]. In Section 7, we show
how to combine the dumbbell tree with topology
trees [10] in order to get a spanner of bounded de-
gree and O(log n) diameter. In Section 8, we show
that spanners that result from the well-separated
pair decomposition have weight O (w[MS2’) log n).
Combining this fact with a result of [2] gives a
t-spanner of weight O ( w(iMST) log n) and diame-
ter O (log n). Finally, in Section 9, we consider all
properties degree, weight and diameter simultane-
ously.
2 Split trees and well-separated
pairs
Virtually all of our spanner constructions will rely
on the notion of a split tree and a well-separated
pair decomposition of a set of points [4, 13, 14]. In
this section, we review these data structures.
A split tree is a tree that stems from a hierarchi-
cal decomposition of a point set into regions that
are k-dimensional rectangles of bounded aspect ra-
tio. There are a number of variants on a split tree.
We outline the fair split tree, due to Callahan and
Kosaraju [4]. Place a smallest-possible k-rectangle
R. about the point set V. The root of the split
tree is R.. Choose the longest side of R. and di-
vide it into two at its bisector. Rectangle R. is
therefore split into two smaller rectangles, RI and
R2. Then the left subtree of R. is the split tree for
RI n V, and the right subtree is the split tree for
Rz n V. The process is repeated until a single point
remains.
In order to simplify some of our arguments, it is
convenient to think of a fair split tree in an ideal
form, which we call the idealized box split tree. In
this tree, rectangles are k-dimensional hypercubes,
each split recursively into 2k identical hypercubes
of half the side length. Actual constructions will be
carried out using the fair split tree., but the ideal-
ized box split tree provides a clean way of concep-
tualizing the fair split tree for purpcmes of analysis.
Next we consider well-separated pair decompo-
sit ions. Let .s >0 be a constant. Two point sets A
and B are well separated if they can. be enclosed in
k-spheres of radius T, whose distance of closest ap-
491
preach is at least sr. A well-separated pair decom-
position is a set of pairs of nonempty subsets of S,
{{ AI, BI}, {AZ, BZ},. . . . {An, Bin}}, such that (1)
the sets Ai and lli are disjoint, (2) for each pair
a, b ~ S, there is a unique pair {Ai, .Bi} such that
a E Ai and b c l?;, and (3) Ai and I?i are well-
separated. Callahan and Kosaraju use a split tree
to compute a set of O(s~n) well-separated pairs in
O(nlog n + s~n) time.
Given these well-separated pairs, Callahan and
Kosaraju show that a spanner can be constructed
easily. For each pair {A~, B;} in the well-separated
pair decomposition, choose arbitrary points, called
representatives, ai E Ai and bi < B;, and connect
a; and bi with an edge in the spanner. Similar
constructions were previously given by Vaidya[14]
and Salowe[13].
3 The dumbbell tree
One of the major difficulties in establishing the re-
sults of this paper is the lack of structure in well-
separated pair decompositions and the spanners
that are derived from them. Unlike the split tree,
well-separated pair decompositions do not possess
any obvious hierarchical structure. One of the ma-
jor innovations of this paper is the observation that
well-separated pair decompositions, and hence the
spanners derived from them, can be decomposed
into a constant number of hierarchically organized
structures. This greatly simplifies the analysis and
construction of spanners, by reducing problems on
general graphs to much simpler problems on trees.
This decomposition may have applications to a
number of other problems where sparse geometric
graphs are used.
Space does not permit a complete presentation
of the decomposition, but the intuition is rela-
tively straightforward. First observe that each
well-separated pair {Al, Bi} can be viewed as a ge-
ometrical object, consisting of two rectangles con-
t aining Ai and Bi, respectively, joined by a line
segment. The resulting shape, is called a durnb-
belt and the rectangles (or in fact, small perturba-
tions of these rectangles) are called the heads of the
dumbbell. The length of a dumbbell is defined as
the distance between the centers of its heads. The
size of a head is defined to be half its diameter.
Das, Heffernan and Narasimhan [7] introduced
the concept of the dumbbell. We claim that it is
possible to partition the set of dumbbells arising
from the well-separated pair decomposition into a
constant number of groups, such that within each
group, dumbbell heads are either disjoint, or one
dumbbell is nested entirely within the head of the
other dumbbell. In particular, we can show the
following (proofs will appear in the full paper):
Theorem 1 Consider the dumbbells resulting
from a well-separated pair decomposition of a set
of n points in dimension k with separation factor
s. In O(n) time it is possible to partition these
dumbbells into O(s)k classes, such that within each
class:
(1)
(2)
(3)
two dumbbells either have lengths that are
within a factor of 2 of one anotherj or else
they differ by a factor of at least s,
any two dumbbells within the same length
interval [x, 2x], are separated by a distance
greater than 2x/s, and
we may deform the heads of each dumbbell
(forming pseudo-dumbbells) such that a dumb-
bell of length x has a head of size at most 4x/s,
and such that the heads of any two pseudo-
dumbbells are either disjoint or else one is
nested within a head of the other.
The nesting of dumbbells provides us with a tree
structure, which we call a dumbbell tree. The im-
portant fact about the dumbbell tree decomposi-
tion is that spanners can be derived from the well-
separated pair decomposition which inherit this
structure. Thus, they can be viewed as consisting
of the union of a constant number of trees. Further-
more, we show that each spanner path is mapped
entirely to one tree. Our main result is summarized
in the following theorem:
Theorem 2 Given a set V of n points in dimen-
sion k, and given t >1, a forest consisting of O(1)
rooted binary trees can be built in O (n log n) time
and O(n) space, having the following properties:
(1) For each tree in the forest, there is a 1-1 cor-
respondence between the leaves of this tree and
the points of V.
492
(2)
(3)
The
Each internal node has a unique representative
point, which can be selected arbitrarily from
the points in any of its descendent leaves.
Given any two points u, v e V, there is a
tree T of the forest, so that the path formed
by walking from representative to representa-
tive along the unique path in T between these
nodes, is a t-spanner path for u and v.
constant factors for the number of trees, pre-
processing ti(e and space are O (Sk log-s), whe;e .s
is O(k/(t – l)). With the addition of an augment-
ing data structure of size O(n), we can compute a t-
spanner path between any two points in O(p+log n)
time, where p is the number of edges on the path.
4 Spanners of bounded degree
In this section, we prove the following general re-
sult, which will be used to construct in O(n log n)
time a t-spanner of bounded degree.
Theorem 3 Let V be a set of n points in IRk
and let t’ > t > 1. Let G be a t-spanner for
V and assume that the edges of G can be di-
rected such that each point has outdegree at most a,
In O(n log n) time, we can construct a t’-spanner
for V in which each point has degree bounded by
O(a (et/(t’ – t))k-l), for some constant c.
In order to prove this result, we need the notion
of single-sink spanner. Let V be a set of points in
IRk, let z be a point of V, and let t >1. A directed
graph having the points of V as its vertices is called
an x-single-sink t-spanner for V if for every point
y in V there is a t-spanner path from y to z.
Let 8 be a fixed angle such that O < 6 < 7r/4
and l/(cos 0 – sin 19) ~ t. Let C be a collection
of k-dimensional cones such that (i) each cone has
its apex at the origin, (ii) each cone has angular
diameter at most d, and (iii) the union of these
cones covers IRk. For each point p 6 IRk and C c C,
let C’Pbethe cone C+p:={a+p:a CC’}.
Now consider the set V and the point z. Let n
be the size of V. For each C G C, let Vc be the
set of all points of V \ {z} that are contained in
the cone CZ. If a point is cent ained in more than
one cone, then we put it in only one subset. If a
subset Vc cent ains more than n/2 points, then we
partition it (arbitrarily) into two subsets Vc,l and
Vc,z, each of size at most n/2.
The z-single-sink t-spanner for V is obtained as
follows. For each subset VC—or in case this set
cent ains more than n/2 points, for each subset Vc,i,
i = 1, 2—we take a point y in this subset that is
closest to x, and we add an edge from y to z. Then
we recursively construct a y-single-sink t-spanner
for this subset. The recursion stops if a subset has
size one,
Using exactly the same analysis as in [12], it fol-
lows that the graph is a single-sink t-spanner.
Lemma 1 Let V be a set of n points in Etk, let
x E V, and let t > 1. In O(nlogn)l timej we can
construct an x-single-sink t-spanner for V, such
that each point has outdegree at most 1 and indegree
bounded by O((c/(t – l))k-l),for some constant c.
Now we are ready to give the transformation that
will prove Theorem 3. Let V be a set of n points
in IRk and let t’> t > 1. Let G be ii, t-spanner for
V and assume that the edges of G can be directed
such that each point has out degree :at most a. We
denote this directed version of G by ~.
For each point x of V, we do the following. Con-
sider all points of V that have an edge in ~ towards
x. Let W be the set of these points. We replace
all edges from W to x by an z-single-sink (t’/t)-
spanner for the set W U {x}.
This gives a directed graph do. We remove
the direction from each edge and call the result-
ing graph Go. We claim that Go is a t’-spanner for
V in which each point has a degree bounded by a
constant.
To prove this, let p and q be any two points of V.
There is a t-spanner path p = PO, PI, P2, . . . . pm = q
in G between p and q. Consider any edg~ {pi, pi+l}
on this path. Assume w.1.o.g. that in G this edge
is directed from pi to p;+l. The directed graph Go
contains a p;+l-single-sink (t’/t)-spanner with p; as
one of its vertices. Hence, in the graph Go there
is a (t’/t)-spanner path between pi and p;~l. The
concatenation of all these paths has length at most
-Z:;l(t’/O lPiPi+ll < (t’/t) t Ipql : t’ Ipql.
Consider the directed graphs G and do. It fol-
lows from Lemma 1 that the outdegrees of both
these graphs are the same. Hence, each point
in Go has outdegree at most a. Let x be any
493
point of V. We bound the indegree of x in co.
This graph contains an z-single-sink spanner hav-
ing ~((~) ~-1) = o((fi)~-l) edges with sink
z. Now let y be any point such that G contains an
edge from z to y. (There are at most Q such points
y,) Then x occurs in a y-single-sink spanner, and it
has indegree bounded by O((ct/(t’ – t))~-l ) in this
spanner. Hence, in the directed graph do, point z
has indegree bounded by 0((1 + a) (et/(t’– t))~-l ).
This proves that in the undirected f-spanner Go,
each point has a degree bounded by a constant.
This proves Theorem 3. It turns out that several
known spanners have the property that their edges
can be directed such that each point has bounded
outdegree. For example, for any O < @ < ~/4, the
9-graph (see [12, 2]) is a t-spanner fort ~ l/(cos 0–
sin t?). This spanner is directed already and each
point has outdegree bounded by 0((c/6)~-1). It
can be constructed in O(n log~–l n) time.
Spanners based on well-separated pair decom-
positions also have the property we need. Essen-
tially, the construction is to enumerate O(n) sets of
“box pairs .“ For each well-separated pair of boxes
{A, B}, choose an arbitrary point a ~ A and b G B
and add an edge {a, b] to the spanner. This edge
is directed from a to b if the parent box of A is not
larger than the parent box of B, then the result-
ing graph has bounded outdegree. (For details, see
Callahan and Kosaraju [5].) The entire graph can
be constructed in O(n log n) time.
Now we can prove the main result of this sec-
tion. Let V be any set of n points in 11%~and let
to > 1. To construct a to-spanner for V having
bounded degree, we set t = & and t’ = to. In
O(n log n) time, we construct a t-spanner G sat-
isfying the condition of Lemma 3. Then we apply
the given transformation and obtain the desired to-
spanner. This proves:
Theorem 4 Let V be a set of n points in ELk and
let t > 1. In O(n log n) time, we can construct a
t-spanner for V in which each point has a degree
that is bounded by a constant only depending on t
and k.
5 Spanners of low weight
In this section, we give an O(n log n) time construc-
tion of a t-spanner that has weight 0( w(JIST)). In
order to bound the weight of this graph, we use a
theorem from Das, Narasimhan and Salowe[9].
Let c >0 be a constant, let A be a set of edges,
and let e G A be an edge of weight 1. If it is possible
to place a cylinder 1? of radius and height c. 1 each,
such that the axis of B is a subedge of e and B n(A\ {e}) = 0, then e is said to be isolated. The set
A has the isolation property if all edges are isolated.
Theorem 5 ([9]) If A has the isolation property,
then w(A) = O(W(MST)), where MST is a mini-
mum spanning tree with respect to the endpoints of
A.
It is easy to see that in the definition of the iso-
lation property, one can replace the cylinder with a
sphere, box, etc., without affecting the above the-
orem.
The low-weight spanner is constructed in the fol-
lowing way. Let C be a cone, and let E(C) be the
set of edges in the box well-separated pair construc-
tion, that, when translated such that one of their
endpoints coincide with the apex of C, lie inside
of C’. We change the endpoints of an edge to en-
sure that the edge does not intersect the interior of
the convex hull of the points within the respective
boxes. These endpoints are chosen from among the
points with maximum or minimum coordinates in a
particular dimension. For each point p, mark edge
e E E(C) if it is the shortest edge in E(C) with
one endpoint in a box ancestor of p. The spanner
G1 consists of the union of the marked edges.
We claim that G1 is a spanner and that its edges
satisfy the isolation property. The fact that G1 is
a spanner can be proved by a straightforward in-
duction proof. To show that G1 has the isolation
property, we use some of the pruning techniques of
Das, Heffernan, and Narasimhan[7]. We may as-
sume that edges have been placed into a constant
number of groups so that each edge has either ap-
proximately the same length or differs in length by
a sufficiently large amount (but bounded by a con-
st ant factor).
We now show that edge e = (a, b) has the iso-
lation property. Edge e corresponds to some well-
separated pair, say {A, B}, in the idealized box
split tree. Note that the length w(e) of e is re-
lated by a constant factor to the diameter d of
these boxes. Place a box @ of diameter d about
494
the center point of e; we first claim that ~ does not
cent ain any points.
To show this, edge e is present because there is
some point p < A, say, for which (a, b) is a shortest
well-separated pair in direction C. If there was a
point q in ~, this would imply that p and q would
be in a well-separated pair. However, the length
of this well-separated pair would be smaller than
w(e), and it would be in the direction of C. (We
note that this new well-separated pair edge may
be in a nearby cone as well; this detail can be fixed
using some pruning techniques. )
We now claim that at most a constant number
of edges intersect a slightly-shrunken version of /3.
Suppose that an edge that is significantly shorter
than w(e) intersects /3. Then we can shrink ~ by
a small amount. Note there are no points inside of
~, so ,0 will not be shrunken by more than a small
percentage.
Suppose that an edge e’ = (a’, 6’) that is sig-
nificantly longer than w(e) intersects @. Then, if
the idealized box A’ containing a’ is sufficiently far
away, any point in A’ would be in a well-separated
pair with a, and the edge corresponding to this
well-separated pair would be shorter than w(e’)
and in approximately the same direction. (Again,
the proof is only sketched. Note that this is where
we need to choose the box representatives in a care-
ful way.)
Finally, consider an edge that has approximately
the same length as w(e). Then these edges must
correspond to idealized boxes within distance w(e)
of A or B. Packing arguments that use the fact that
w(e) is related to the width of A imply that there
are only a constant number of idealized boxes in
this area. Therefore, there are only at most a con-
stant number of edges that can intersect ~. Again,
using the decomposition technique of Das et al. [7],
one can partition the group of edges into a constant
number of edge sets, each possessing the isolation
property. The following theorem is proved.
Theorem 6 In any dimension, jor any t > 1, t-
spanner G1 can be constructed in O (n log n) time,
and it has weight O(W(MST)).
Our construction actually has bounded degree as
well. This is because any set of edges possessing the
isolation property has bounded degree (a straight-
forward proof ). We therefore have the following:
Corollary 6.1 In any dimension and for any t >
1, t-spanner GI can be constructed in O(n log n)
time, and it has weight O(W(MST)) and bounded
degree.
6 Spanners of small diameter
We first consider the case of the l-dimensional
spanner, and then we show that a,ll the higher-
dimensional cases are closely related to the 1-
dimensional case.
In the l-dimensional case, the input is a set of
n points on a line, and the output is a graph with
small diameter. Surprisingly, a useful construction
has already been discovered. It was devised by
Alon and Schieber[l]. Among other results, this
construction implies that there is a linear-sized 1-
spanner with diameter a(n) + 2.
Here are the essential aspects of the Alon and
Schieber construction (they are tailcmed somewhat
to enhance the analogy to our probllem). Suppose
we want a spanner of diameter d that contains as
few edges as possible. Alon and Schieber divide up
the point set into 1 pieces, each piece of size n/4.
For each piece, recursively construct a spanner of
diameter d; this accounts for spanner paths within
the pieces. In order to account fclr the spanner
paths between the pieces, select the points in each
group with smallest and largest values. Each of the
points in a particular group are connected directly
with the two group representatives; the represen-
tatives themselves are connected with a spanner of
diameter d – 2.
The number of edges T(n, d) used in the Alon
and Schieber construction is given by the recur-
rence:
T(n, d) = O(n) -t
T(n, 1) = 0(n2).
By choosing the values
T(21, d – 2) + fI’(n/(, d)
of 4 appropriately, it is pos-
sible to show that T(n, 2) = O(nlog n), T(n, 3) =
O(nloglog n), T(n,4) = O(nlog” n), and so on. It
is also possible to show that the diameter is a(n)+2
if one allows only O(n) edges.
In order to generalize this idea to the k-
dimensional case, we use the fact that there exists
a spanner which can be represented as the union of
a constant number of bounded degree trees. (See
495
Theorem 2.) Let T be one of these trees. We
construct a modified version of T whose degree
is bounded by a constant, and we endow it with
some additional geometric properties. Specifically,
this modified dumbbell tree T’ is a tree whose ver-
tices are original points and whose edges are Eu-
clidean edges. The import ant geometric property
of T’ is the following: if a pair of points a and b are
in a well-separated pair that actually appears as a
dumbbell in T, then the path between a and b in
T’ is a t-spanner path.
The actual construction of T’ is done in the fol-
lowing way. A dumbbell A in T contains several
children dumbbells Al, Az, . . . . Am and several iso-
lated points PI, p2, . . .pj. Consider these isolated
points to be degenerate dumbbells and therefore
children of A. This possibly large set of edges will
be replaced by a tree T’” whose degree is bounded
by a constant, described below.
For each box in Ai, choose a representative
point. Let tree T“ be the minimal tree (with re-
spect to edge inclusion) in the fair-split tree that
connects these representative points. This tree T“
is a Steiner tree: it consists of original points (the
represent ative points), Steiner points (degree-3 ver-
tices), and paths connecting these two types of
points. T’” is the tree that results from replacing
each of the paths in Tti with a single edge.
The proof that T’ has the t-spanner path prop-
erty stems from the fact that the children dumb-
bells are much smaller than the parent dumbbell
and the fact that the diameter of a box is halved
in the fair-split tree within a constant number of
levels. A detailed proof is omitted.
We apply a construction akin to the one of
Alon and Schieber to shortcut the paths. Let
P = Z1, X2,. . .,aj be a path in T’. By the trian-
gle inequality, any path P’ = X1, ZP(2), ZP(3), . . . . xj,
where 1 < p(2) < p(3) < . . . < j has length
less than or equal to the length of P. Appropri-
ate shortcuts, therefore, have spanner properties.
The details of how these shortcuts are constructed
is omitted.
Theorem 7 For any t >1, and any dimension k,
there is a t-spanner containing O (n) edges and con-
structible in O(n log n) time with diameter a(n) +2.
If one allows more space, the diameter can be
reduced. Here are some of our results.
Theorem 8 For any t >1, and any dimension k,
1.
2
3.
7
there is a t-spanner containing O(n log n)
edges and constructible in O (n log n) time with
diameter 2,
there is a t-spanner containing O(n log log n)
edges and constructible in O (n log n) time with
diameter 3,
there is a t-spanner containing O(n log* n)
edges and constructible in O (n log n) time with
diameter 4.
Spanners of bounded degree
and small diameter
Theorem 9 For any t > 1, and any dimension
k, in O(n log n) time, a t-spanner whose degree is
bounded by a constant and whose diameter is at
most O (log n) can be constructed.
Our low-diameter constructions of the previous
section have high degree. On the other hand, it is
difficult to bound the diameter of our bounded-
degree constructions. Note, however, that our
diameter results are closely related to the one-
dimensional results.
Consider the following strategy to produce a
spanner of O (log n) diameter and bounded degree
for a set of n points on a horizontal line. Without
loss of generality, assume that n is a power of 2 and
that they are numbered O through n – 1 from left
to right.
Include an edge (i, i + 1), O ~ i < n. The result-
ing graph is a spanner, but its diameter is n – 1.
Select the set of even-numbered points, 0,2,4,...,
and connect them by a set of edges, (2i, 2i + 2),
O ~ i < n/2. Repeat this process. The resulting
set of edges has log n diameter, but several of the
points have degree log n as well.
In order to reduce the degree, note that O (log n)
diameter would have been preserved if the odd-
numbered points were chosen at the second “level,”
or if either 2i or 2i+ 1 was “promoted” to the second
level. A similar statement can be made at the l-th
level (2~i through 2t(i + 1) – 1 can be promoted). If
one is careful about alternating “promotions ,“ the
496
resulting structure, reminiscent of a bounded de-
gree skip-list, has bounded degree and logarithmic
diameter.
In order to generalize this proof to all dimen-
sions, we need to apply the same strategy to trees,
specifically the modified dumbbell tree of Section
6. Here, the appropriate analogue to the leveling
idea seems to be Frederickson’s topology trees[lO].
We provide a rough outline of the method and
the properties we need to maintain. Suppose we
have a rooted tree T whose degree is bounded by
a constant. Furthermore, assume that every leaf
node has a unique label and that any internal node
can be labeled with the label of an arbitrary leaf
node. The first step is to choose representatives
for the nodes in T. To do this, we propagate leaf
labels. A node chooses one of the propagated labels
and propagates the other up the tree. Each label
is used at most twice, once at a leaf, and once at
an internal node.
We then perform a layering approach, grouping
sets of nodes into a single node at the next layer.
An important issue is the maintenance of a tree
whose maximum degree is bounded by a constant
at every level.
Given this layered tree, labels are again dis-
tributed so that no label is used more than a con-
stant number of times. Roughly, the labeling pro-
cedure ensures that points (corresponding to the
labels) have degree bounded by a constant, and
the leveling process ensures that the path has link-
distance O (log n). Full details will be included in
the final paper.
8 Spanners of low weight and
small diameter
We use the following spanner construction, due to
Arya, Mount, and Smid[2]: Start with a fair-split
tree, and designate some nodes as heavy and some
as light. A node is heavy if it cent ains more points
in its subtree than its sibling, and it is light other-
wise (if both subtrees contain an equal number of
points, the left child is heavy and the right child
is light). We use this designation to determine box
representatives for the well-separated pairs; specifi-
cally, a parent box inherits the representative of its
heavy child. Arya et al. [2] show that if the repre-
sentatives are chosen in this way, then the resulting
spanner has diameter O (log n).
We now show that the weight of well-separated
pair constructions is O(ZO(MST) log n), which is
tight [11]. This improves the results of Lenhof et
al. [11], who prove that the sum D of the diameters
of the boxes in a box split tree is O(W(MST) log2 n)
and that the length of the well-separated pair edges
is O(D). Our techniques can be used to show that
D = O(w(MST)logn).
Rather than focus on the split tree, we focus
on the dumbbell tree. Recall the gap property
of Chandra et al. [6]: A set of edges E has the
gap property if for every pair el and e2, the dis-
tance between the closest endpoints of el and e2”
is at least the length of the smaller edge. Chan-
dra et al. prove that if E has the gap property,
W(E) = O(w(iwsz’)logn).
In our case, let E be the set of well-separated
edges represented by a dumbbell tree. We show
that there is a set of edges El ~ E such that El
has the gap property and w(E’) = El(w(E)). This
proves that w(E) = O(W(MST) Iogn).
To select E’, initially let E’ = E, and consider
any pair of edges el and e2 in E’. l[f they violate
the gap property, remove the shorter one, say el,
and continue with E’ \ {el }. Eventually, E’ will
have the gap property.
In order to show that w(E’) = @(w(E)), build
the following directed forest: when e-l is eliminated
because of e2, direct an edge from e2 to el. Note
that only the root e of a tree t(e) in the forest
will remain in E’, so we want to show that w(e) =
@(w(t(e))).
Consider the children of edge e’ in t(e). Re-
call the length grouping property c~f Theorem 1.
The children of e’ are of length i~pproximat ely
c%. w(e’), where i > 0 indicates the length group,
and O < c << 1 is a constant, From dumbbell tree
properties, only a constant number & of edges of
length 1 may be within a distance 1 of a fixed point,
so the number of children of e’ in group i is at most
2(, f for each endpoint. The parameter c can be
chosen independently off, so that ~c < 1/2. It fol-
lows that the total weight of the children of e’ is at
most fi~w(e’). This implies that w(t(e)) s ~,
where 6 = &~, which in turn implies the main
theorem in this section,
497
Theorem 10 For any t > 1, and any dimen-
sion k, there is a t-spanner, constructible in
O(n log n) time, with O(log n) diameter and weight
o(w(MsT)logn).
9 Spanners of bounded degree,
low weight and small diameter
It turns out that our bounded degree, O(log n)
diameter spanner also possesses some interest-
ing weight properties. Our analysis above shows
that the sum of the diameters D of the boxes
in an appropriate box split tree construction is
0( w(MST) log n), so one layer of the construc-
tion has weight O ( W( MST) log n). Since there are
O(log n) layers, the weight is O(W(MSZ’) log2 n).
We conclude with the following result:
Theorem 11 For any t > 1, and any dimension
k, there is a t-spanner, constructible in O(n log n)
time, with bounded degree, O(log n) diameter, and
weight O(W(MSZ’) log2 n).
Conjecture 1 For any t >1, and any dimension
k, there is a t-spanner, constructible in O(n log n)
time, with bounded degree, O (log n) diameter, and
weight O(W(MSZ’) log n).
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